Ultra-Narrowband Silicon-Micromachined Sub-THz Filter With Wide Spurious-Free Rejection Band Employing High-Q TM330 Resonators

In this article, we present an ultra-narrowband silicon-micromachined bandpass filter with a wide and high-rejection stopband. The proposed filter uses high-<inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> factor TM330 mode resonators. To avoid near-passband spurious resonances typically associated with higher order modes, a novel method of arranging the positions of the coupling slots is carried out. A fourth-order filter with a center frequency of 183 GHz and a fractional bandwidth (FBW) of 0.5% has been fabricated by silicon micromachining for the first time. The prototype employs out-of-plane transitions on the input and output ports, which results in axial ports enabling a direct characterization with the device simply mounted between the two standard waveguide test ports. The unloaded <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> factor extracted from the measurements is 1000, which is unparalleled by any previously reported narrowband filter in any technology in this frequency range. A spurious free response with a high stopband rejection in the entire waveguide band is obtained. The measured worst-case insertion loss and return loss (RL) in the passband are 4.5 and 9 dB, respectively.


I. INTRODUCTION
I N MODERN communication systems, low-insertion-loss narrowband filters with a wide spurious-free high-rejection stopband are highly desired, especially in sub-THz and THz communication systems, where propagation losses and spectral efficiency play a critical role [1], [2], [3].High-quality factors (Q factors) of waveguide resonators make waveguide filters the most suitable choice for low-loss applications [4].The cavities' Q factor at sub-THz and THz frequencies is dominated by the surface roughness and ohmic losses of the metallization, making the design of low-loss filters challenging at high frequencies [5].The authors are with the Division of Micro-and Nanosystems, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: mohmg@kth.se;glubokov@kth.se;joachimo@kth.se).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TMTT.2023.3326287.
Selection of the type of resonators and their resonant modes is an important step in the filter design.Typically, in rectangular waveguide filters, the fundamental mode is chosen due to the compact size of the cavities; in this case, the stopband is usually spurious free, as there is no resonance in the lower stopband, and normally, the upper stopband resonances have an acceptable distance from the passband [24], [25], [26].At the same time, the fundamental mode cavities exhibit lower Q factors than higher order mode cavities.In addition, higher order mode cavities are less sensitive to fabrication tolerances; the latter property is particularly important for THz frequencies when cavities are extremely small [18], [27], [28].
However, the stopband performance of the higher order mode filters is severely affected by spurious modes occurring in the vicinity of the passband.Therefore, it is necessary to improve the out-of-band rejection during the filter design process.Various methods have been applied to enhance the stopband performance: 1) separation or pushing spurious resonances away from the passband by changing resonator's geometry or modifying the structure based on the field distribution [27], [29], [30]; 2) suppression of undesired modes by altering coupling structures, considering the electromagnetic field distribution [24], [31]; 3) generation of transmission zeros [27], [32], [33].Many of these previous studies have focused on suppressing just one or two upper band spurious modes, especially in substrate-integrated waveguide (SIW) filters that offer relatively simple fabrication processes and easy manipulation of the structure [24], [28], [29], [30], [31], [33].In addition, in [29], the out-of-band enhancement was carried out without considering that the method also drastically reduces the Q factor.
This study proposes a method of designing an ultra-narrowband bandpass filter with low insertion loss and a wide and high-rejection stopband at sub-THz frequencies.For the first time at sub-THz frequencies, a filter with fractional bandwidth (FBW) as low as 0.5% is successfully realized through silicon micromachining technology.The filter employs high-Q TM 330 square cavities arranged in a zigzag topology with external and internal double-slot coupling to efficiently suppress spurious resonances within the entire WR-5 band (140-220 GHz).The fourth-order prototype filter's cavities show the highest extracted Q factors reported in the WR-5 band so far.

II. FILTER DESIGN
The design specifications of the fourth-order narrowband bandpass prototype filter are as follows.

A. High-Order Mode Cavity
Fig. 1 shows the magnetic field distribution in cavities with the fundamental (TM 110 ) and the TM 330 modes designed to occur at the same resonant frequency.The Q factor of a resonator is defined by the losses per period.For air-filled rectangular waveguide cavities, the losses mostly occur due to the conductive walls.The unloaded Q factor of TM mn0 resonant modes of a rectangular waveguide cavity is expressed as follows: where a, d, and h are the rectangular cavity's length, width, and height, respectively; R s = (ωµ/2σ ) 1/2 is the surface Fig. 2. Ratio of higher order TM modes' Q factors to the fundamental TM mode's Q factor of square cavities with respect to their heights, multiplied by their resonance frequencies.
resistivity of the conductive walls; and f 0 is the resonant frequency.A rectangular cavity's resonance frequency can be determined by the following expression: ( For a constant resonant frequency for different modes ( f 0 = f mn0 ), using ( 2), (1) transforms to where According to (3) In order to achieve the highest Q factor, we consider square cavities (a = d) with symmetric modes (m = n) Fig. 2 shows the ratio of Q factors of higher order symmetrical modes (TM mn0 ) to the fundamental mode (TM 110 ) for square cavities with respect to the cavity height multiplied by the resonant frequency.This graph serves as a reference for selecting a high-Q factor resonant mode based on a desired operational frequency and for height restrictions imposed by the fabrication process while keeping the structure reasonably compact.In general, higher order modes provide larger Q factors at the expense of size, especially for large cavity heights or high frequencies.However, as the order of the mode increases, the rate of growth decreases.The TM 330 mode is chosen for the prototype filter, since larger cavities unnecessarily exceed the structure size while not further improving the Q factor significantly.Therefore, the size of the cavities is chosen, such that their TM 330 mode resonance appears at the selected center frequency of the filter (using ( 2), a = d = 3477 µm).Compared with the first mode at the chosen center frequency (183 GHz), and for the chosen cavity height of 556 µm, determined by the double height of the silicon wafer thickness, the TM 330 provides about 50% higher Q factor.The Q factor is high enough to design a narrow BW filter with an FBW of 0.5%.

B. Spurious Resonances Suppression
Filters with higher order mode resonators are characterized by spurious resonances occurring close to the passband.The employed TM 330 square cavity has eight potential spurious modes in the WR-5 band.These modes are TM 230 , TM 320 , TM 410 , TM 140 , TM 340 , TM 430 , TM 150 , and TM 510 .Depending on their design and configuration, conventional iris couplings can suppress only some of these unwanted TM mn0 modes.Fig. 3 shows two examples of fourth-order direct-coupled filters utilizing iris-coupled TM 330 cavities: an inline and a folded design.The simulated responses show several unwanted spurious resonances in the upper and in the lower stopbands.Specifically, Jaschke et al. [30] reported the topology from example 2 in Fig. 3, where an unwanted second-order mode was efficiently suppressed by shifting the positions of coupling irises; however, this method cannot be applied for the TM 330 mode cavities in the proposed design, since the filter does not use irises in resonators' side walls as coupling structures.
The configuration of the proposed fourth-order bandpass filter for the prototype in this article is shown in Fig. 4. The filter is designed in two layers of cavities, which are arranged in a zigzag topology.This topology was selected to minimize the required number of layers for the structure, resulting in easier fabrication and reduced manufacturing costs.Also, the zigzag topology enables implementation of all internal and external couplings by means of double slots, allowing efficient suppression of unwanted modes, which would not be possible with irises.Furthermore, zigzag topology is free from irises, which are subjected to underetching and introduce extra fabrication uncertainty.The slots are fabricated in the thin device layer, so the underetching is negligible and leads to higher accuracy of the couplings.The second-row cavities are shifted by third of the cavity length (b) to create space for coupling slots between resonators 2 and 3.
Fig. 5 shows the simulated magnetic field distribution of the device.All the couplings are implemented using two parallel slots located at the magnetic field maxima of TM 330 to obtain sufficiently high coupling values with small apertures, to minimize Q-factor sacrifice.The rejection band is improved significantly by employing two slots for each coupling structure, placing them so that the magnetic field direction is in line with only the TM 330 mode.This prevents the excitation of TM 140,410 , TM 150,510 , TM 230 , and TM 430 modes from the TM 330 electromagnetic field, since the direction of the magnetic field of these modes at coupling slots is not inline with TM 330 resonance mode (which yield to zero coupling between TM 330 mode and the spurious modes).Moreover, the feed is designed, such that the electromagnetic wave of the rectangular waveguide also couples to the cavity through the two slots positioned, such that the field directions in the waveguide and the TM 330 mode of the cavity are aligned.Therefore, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Furthermore, the minima of the magnetic fields of modes TM 320 and TM 340 occur at the slots; thus, these modes are also suppressed.As a matter of fact, the coupling network allows the TM 130 mode or some very high-order modes to couple through slots, since their magnetic fields are in line with their resonant frequencies; however, they are out of the band of interest.
The red rectangles in Fig. 5(a) show the locations of external coupling slots connecting the input-output rectangular waveguides and cavities 1 and 4. The slots are located on the roof of cavity 1 and at the bottom of cavity 4. The internal coupling slots between resonators 1 and 2 (M 21 and M 12 ), as well as between resonators 3 and 4 (M 43 and M 34 ), are displayed as blue rectangles in Fig. 5(a).The two narrower green rectangles in Fig. 5(a) display the coupling slots between resonators 2 and 3 (M 32 and M 23 ).
The internal and external couplings are adjusted by changing the size of the slots.Fig. 6 shows how the coupling coefficient changes with adjustment of the size of the coupling slots.It is evident that enlarging the slot size results in increased coupling.The internal coupling coefficients and external quality factors are determined using ( 6) and ( 7), as in [34] For the internal couplings, the resonance frequencies f 1 and f 2 are determined from the simulated results of two TM 330 cavities stacked on top of each other with a shift (such as in the zigzag configuration used in the design) and connected with two identical slots; the procedure is performed under the low external coupling condition.For the external Q factor, f 0 is the resonance frequency, and τ is the group delay calculated from the reflection coefficient of a single resonator connected to the feeding circuit through the double slot.I shows the final dimensions of proposed filter after optimization.The sizes of cavities 1 and 4 are slightly different from cavities 2 and 3 due to loading effects.

C. Feeding Circuit
In the majority of previous works, silicon-micromachined waveguide devices were characterized using CNC-milled splitblock fixtures providing interfaces between the ports of the filters and standard waveguide flanges [4], [35], [36], [37].As the authors have shown previously, an on-chip axial port interface enables more accurate measurements by minimizing potential misalignments [38], fabrication tolerances, metal surface roughness, and allows the chip to be inserted directly between waveguide flanges, i.e., requires no measurement fixtures [39].Thus, the filter can be characterized by employing two feeding circuits in layers 1 and 6 to provide axial port configuration.The circuits consist of two bends: an E-plane bend that enables transition from waveguide port to the micromachined in-plane waveguide and an H -plane bend that rotates the in-plane waveguide by 90 • .The configuration of the feeding circuit is shown in Fig. 7.The step and the shorted waveguide section are used to enhance the E-plane bend's matching.
The transition design is optimized using CST Microwave Studio to achieve better than 20 dB RL for 163-210 GHz.In comparison, previously reported micromachined transitions requiring several wafer layers [39], [40], [41], [42], the proposed transition is designed to be fabricated using a single silicon-on-insulator (SOI) wafer, significantly reducing fabrication complexity and the size of the structure.

III. TOLERANCE ANALYSIS
DRIE enables high aspect ratio anisotropic etching, but also causes underetching, i.e., nonverticality of the side walls, for very deep etching steps, since the rate of physical etching decreases with increasing etching depth.However, such underetching is not problematic, since it is well reproducible and, thus, can be measured after a test fabrication run and compensated by the design [5], [20], [23], [43].
Nonvertical sidewalls have an influence on the coupling performance of in-plane waveguide irises but have no impact on coupling structures implemented in the roof or the bottom of the cavities, since underetching for thin layers is negligible.Fig. 8(a) shows a sensitivity analysis illustrating the impact of the nonvertical sidewalls on the performance of the proposed filter.The graph shows that the underetching has a negligible effect on the coupling slots, as anticipated; however, due to the size expansion of the cavities, the center frequency of the filter shifted.Since the higher order mode resonators are less sensitive to dimensional tolerances, the center frequency changes only by 0.5% for 20-µm underetching.Underetching has been considered during the design process; the cavities are designed with 20-µm underetching, and the feeding circuit with 4-µm underetching, based on previous fabrication, experiences.
The second factor that may degrade the performance of the multilayer filter is bonding misalignment.A sensitivity analysis with 20 uniformly distributed random misalignment values within ±10 µm in the xand y-directions is shown in Fig. 8(b).The design shows low sensitivity to assembly tolerances within practical previously observed values.
Silicon micromachining based on DRIE is a very robust fabrication process and has proven very low-dimension tolerances (lower than 1 µm, which mostly comes from the lithography step).Fig. 8(c) shows the sensitivity analysis of the filter to the variation of its geometrical parameters.In this analysis, all of the etched areas, including size and positions of cavities, slots, and the feeding circuit, have been randomly sampled using the Gaussian distribution with 1-µm standard deviation and centered at the optimal design values.The plot demonstrates that the filter design has low sensitivity to the dimensions tolerances within the values expected from the utilized fabrication technology.

IV. FABRICATION AND ASSEMBLY
The proposed bandpass filter is fabricated based on standard micromachining processes on six SOI layers.Four middle layers are used for the filter body, and the two outside layers are used for the feeding circuit.The SOI wafer consists of two silicon layers: a handle layer (275 µm) and a device layer (30 µm), separated by a thin (3 µm) buried oxide (BOX) layer.The BOX layer is used as an etch-stop layer, resulting in a smooth surface for the bottom of the etching area [6], [9].Fig. 9 shows the fabrication process flow consisting of five major steps: starting with growing 2-µm thermal silicon oxide on both sides of the wafer.Following that, a dry oxide etching process is used to create the mask.Then, both silicon layers are etched through the DRIE process.The cavities have been made by etching the handle layer, and the device layer Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. is etched to create the coupling slots.After removing the oxide layers, the chips are metalized by a 1.5-µm gold layer.A thermocompression method was used to bond the chips [43].

V. EXPERIMENTAL VERIFICATION
The filter is characterized by a Rohde & Schwarz ZVA24 vector network analyzer with two WM-1295 millimeterwave VDI extenders for 140-220-GHz range.Calibration is performed using a standard thru-reflect-line waveguide calibration kit, and the reference planes are defined at the test port adapter flanges of the two frequency extenders.
The measured S-parameters of the fabricated filter are plotted in Fig. 11 along with the simulation results.Although there is a generally good agreement between the measurement and simulation results, the insertion and RLs of the filter are significantly affected.The error analysis indicates that the main source of the poor RL is the postbonding misalignment error of approximately 70 µm between layers 1 and 2 (the misalignment is estimated based on Vernier scales in the corner of the chips, shown in Fig. 12), which significantly affects the performance.This, most likely, happened during transferring the assembled stack to the oven for bonding.The typical maximum misalignment values do not exceed 5 µm (which is also true for misalignment between all the other layers in the presented filter), and the average values on Resimulated response of the filter with applied experimentally determined misalignment values compared with the measured data, together with a microphotograph of the alignment features on the chips.the previous chips fabricated by our group with the same procedure are less than 2 µm [5].Fig. 12 shows the simulated response including the measured experimental misalignment.The experimental and resimulated responses demonstrate quite good agreement; at the same time, the discrepancy can be attributed to underetching, which slightly deviates from the predicted values.
The result shows a wide spurious-free rejection band, which confirms the efficiency of the proposed method.The passband has a minor 0.1% shift to higher frequencies because of an underestimation of the underetching at the design stage, which can be completely compensated by a second fabrication run.A stabbing point with a peak resonance lower than −70 dB occurs at 179 GHz, which is attributed to a small coupling between TM 330 and TM 140 modes, which increases with growing underetching and misalignment between layers, according to the simulated field distribution.This resonance does not affect the response of the filter significantly.The measured worst-case insertion loss and peak RL in the passband are 4.5 and 9 dB, respectively.The unloaded Q factor of 1000 is extracted for a single cavity from the measured S-parameters of the filter using vector fitting [44].
Table II shows the comparison of recently published THz filters' performances.The proposed filter has the lowest FBW among all the filters published so far at sub-THz and THz ranges and shows the best extracted Q factor despite the detuning of the RL due to fabrication errors.To illustrate the performance of the presented filter with respect to ideally tuned lossy prototypes, we analyzed the responses of the prototypes with various unloaded Q factors and FBWs.Fig. 13 demonstrates the variation of insertion losses of an ideally tuned lossy direct-coupled fourth-order filter with 20-dB RL at different Q factors and FBWs.A 0.5% FBW filter with Q factors of 1000 would have had the minimum insertion loss of about 4 dB, provided that it is perfectly tuned, while the presented filter has only 4.5-dB insertion loss because of imperfect tuning due to mainly assembly errors occurred in the complex multilayer structure.

VI. CONCLUSION
An ultra-narrowband fourth-order silicon-micromachined bandpass filter with a wide and high-rejection stopband for sub-THz frequencies has been designed and fabricated.High-Q factor TM 330 mode cavities together with a network of coupling slots have provided low insertion loss and suppressed the spurious resonances associated with higher order modes.The measured performance of the filter has shown a 4.5-dB insertion loss and 9-dB RL and an over 40-dB rejection on the stopband in the entire WR-5 band.The measured results are in good agreement with the numerical predictions.The measured data have shown an excellent unloaded quality factor of 1000.

Manuscript received 15
June 2023; revised 30 August 2023 and 10 October 2023; accepted 12 October 2023.Date of publication 31 October 2023; date of current version 5 June 2024.This work was supported in part by the Swedish Research Council (VR), in part by the European Union's Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Grant Agreement 811232-H2020-MSCA-ITN-2018, and in part by the Swedish Foundation for Strategic Research under Grant Agreement CHI19-0027.(Corresponding author: Mohammad Mehrabi Gohari.)

Fig. 1 .
Fig. 1.Magnetic fields of TM 110 and TM 330 modes having the same resonant frequency.

Fig. 3 .
Fig. 3. Simulated S-parameters of two conventional iris-coupled filters with TM 330 mode cavities, displaying many unwanted spurious resonances.

Fig. 4 .
Fig. 4. Computer aided design (CAD) model of the TM 330 mode filter.(a) 3-D view of the filter with two transitions.(b) Side view, featuring the wave path.(c) Coupling topology and coupling matrix of the filter.

Fig. 5 .
Fig. 5. Magnetic field distribution and positions of transition and coupling slots in the proposed filter.(a) H -plane cross section.(b) Side cross section.

Fig. 6 .
Fig. 6.Internal and external couplings in relation to the slot dimensions.(a) Inter-resonator coupling coefficient when two TM 330 mode cavities are connected through double slots positioned at magnetic field maxima.(b) External Q factor when the feeding structure is connected to a single TM 330 mode cavity through the double slots.

Fig. 4 (
Fig.4(c) presents the coupling matrix of the fourth-order filter.The corresponding values of its entries are as follows:M S1 = M 1S = M L4 = M 4L = 1.035,M 12 = M 21 = M 34 = M 43 = 0.911, and M 23 = M 32 = 0.7.TableIshows the final dimensions of proposed filter after optimization.The sizes of cavities 1 and 4 are slightly different from cavities 2 and 3 due to loading effects.

Fig. 8 .
Fig. 8. Sensitivity analysis of the proposed filter.(a) Underetching effect.(b) Chip assembly misalignment: the gray curves represent the S-parameters of 20 random (normally distributed with a standard deviation of 10 µm in both directions) shifts between the chips in vertical and horizontal directions.(c) Dimensions tolerances: the gray curves represent the S-parameters of ten random designs, where all the dimensions of cavities and slots are normally distributed with a standard deviation of 1 µm from optimal design values.

Fig. 9 .
Fig. 9. Schematic fabrication process flow.(a) SOI wafer coated with 2-µm thermal oxide layers.(b) Hard masks preparation on the SOI wafer by dry-etching silicon oxide.(c) DRIE of the device layer and the handle layer.(d) Oxide dry-etching BOX layer and hard masks.(e) Metallization.(f) Thermocompression bonding.

Fig. 10 .
Fig. 10.Micrograph of the fabricated chip of bandpass filter.(a) Device layer side (top) and handle layer side (bottom).(b) Assembled six-layer chip.

Fig. 10
Fig.10shows the front sides and the backsides of the six layers and the bonded chip.

Fig. 11 .
Fig. 11.Measured and simulated S-parameters of the proposed bandpass filter.(a) Entire WR-5 band.(b) Magnified view of the passband.

Fig. 12 .
Fig. 12.Resimulated response of the filter with applied experimentally determined misalignment values compared with the measured data, together with a microphotograph of the alignment features on the chips.

Fig. 13 .
Fig. 13.Variation of insertion losses of the fourth-order direct-coupled filter prototypes with 20-dB RL with respect to the resonators' Q factors.(a) S-parameters of the filter prototypes with 0.5% FBW.(b) Minimum achievable insertion losses of the filters for various FBWs.

TABLE II PERFORMANCE
COMPARISON OF SUB-THz BANDPASS FILTERS